Parametrically Excited Stability of a Simply Supported Beam Under Axial Periodic Excitation

Abstract

The parametrically excited stability of beams with multiple mode vibration under general periodic axial excitation is studied and the periodic transverse supports are considered for the first time. The partial differential equation of motion of the beam with spaced supports under axial excitation is given and then converted into ordinary differential equations with periodic parameters by using the Galerkin method. The direct eigenvalue analysis method is applied to solve the parametrically excited stability of the beam described by the differential equations with periodic parameters based on the Fourier expansion and generalized eigenvalue analysis. A track beam with spaced supports under periodic axial excitation is considered. Numerical results on unstable regions are given to illustrate the parametrically excited stability of the beam and the influence of supports and excitation on the stability. Introduction The dynamic stability of periodically parametrically excited systems is an important research subject in engineering. For example, the parametrically excited vibration of cables in a cable-stayed bridge under support motion excitation can result in the cable instability and damage [1-3]. The parametrically excited vibration of beams under axial periodic excitation can result in the dynamic buckling [4-5]. The periodically parametrically excited systems can be expressed as the Hill equations or Mathieu equations. The stability of single Mathieu equation representing single-degree-of-freedom system has been quite studied by using the Floquet theory [6]. Several approximation methods for solving the stability of coupled Mathieu equations representing multi-degree-of-freedom system have been proposed. Based on the Floquet theory and harmonic balance method, the direct eigenvalue analysis method for the parametrically excited stability have been developed and applied to inclined stay cables with multi-degree-of-freedom to obtain unstable regions [7-9]. The parametrically excited stability of beams under axial periodic excitation has also been studied [4-5]. However, the stability analysis was based on single or several mode vibration and harmonic axial excitation. The parametrically excited stability of beams with multiple mode vibration under general periodic excitation needs to be studied further, which has a more practical significance in engineering. The present paper mainly focuses on the parametrically excited stability of a beam with coupled multiple mode vibration under general periodic axial excitation, and the periodic supports for improving the beam stability are considered. First, the differential equation of motion of the beam with spaced supports under axial excitation is given. The Galerkin method is applied to convert the partial differential equation into ordinary differential equations. Second, the direct eigenvalue analysis method is applied to solve the parametrically excited stability of the beam with spaced supports under axial excitation described by the differential equations with periodic parameters. Finally, a track beam with spaced supports under periodic axial excitation is considered. Numerical results on unstable regions are given to illustrate the parametrically excited stability of the beam and the influence of supports and excitation on the stability.